1997 PMWC Problems/Problem T3

Problem

To type all the integers from 1 to 1997 using a typewriter on a piece of paper, how many times is the key '9' needed to be pressed?

Solution

Let's call the three digit, two digit, and one digit numbers, when combined, the 0 thousands.

The 0 thousand has the same number of nines as the one thousand, so we can compute the number of nines in the 0 thousands and multiply it by 2 and subtract 5, since we are leaving out 1998 and 1999.

one digit: one nine, obviously.

two digit: a nine times nine, plus 10 other nines, is 19 nines.

three digit: 20 per hundred, plus another hundred for the 900s, is 280.

20+280=300. 300*2-5=595.